23 research outputs found
Elliptic Littlewood identities
We prove analogues for elliptic interpolation functions of Macdonald's
version of the Littlewood identity for (skew) Macdonald polynomials, in the
process developing an interpretation of general elliptic "hypergeometric" sums
as skew interpolation functions. One such analogue has an interpretation as a
"vanishing integral", generalizing a result of arXiv:math/0606204; the
structure of this analogue gives sufficient insight to enable us to conjecture
elliptic versions of most of the other vanishing integrals of
arXiv:math/0606204 as well. We are thus led to formulate ten conjectures, each
of which can be viewed as a multivariate quadratic transformation, and can be
proved in a number of special cases.Comment: 54 pages, LaTeX; v2: main conjectures renumbered, additional
consistency conditions and several more special cases proved. v3: references
to further progress added, various clarification
Aspects of elliptic hypergeometric functions
General elliptic hypergeometric functions are defined by elliptic
hypergeometric integrals. They comprise the elliptic beta integral, elliptic
analogues of the Euler-Gauss hypergeometric function and Selberg integral, as
well as elliptic extensions of many other plain hypergeometric and
-hypergeometric constructions. In particular, the Bailey chain technique,
used for proving Rogers-Ramanujan type identities, has been generalized to
integrals. At the elliptic level it yields a solution of the Yang-Baxter
equation as an integral operator with an elliptic hypergeometric kernel. We
give a brief survey of the developments in this field.Comment: 15 pp., 1 fig., accepted in Proc. of the Conference "The Legacy of
Srinivasa Ramanujan" (Delhi, India, December 2012
Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems
Multidimensional matrix inversions provide a powerful tool for studying
multiple hypergeometric series. In order to extend this technique to elliptic
hypergeometric series, we present three new multidimensional matrix inversions.
As applications, we obtain a new elliptic Jackson summation, as well as
several quadratic, cubic and quartic summation formulas
Elliptic hypergeometric functions associated with root systems
We give a survey of elliptic hypergeometric functions associated with root
systems, comprised of three main parts. The first two form in essence an
annotated table of the main evaluation and transformation formulas for elliptic
hypergeometric integeral and series on root systems. The third and final part
gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part
also developed by Coskun and Gustafson).Comment: 28 pages. Modulo minor edits this paper will appear as a chapter in
the book "Multivariable Special Functions" (edited by Tom Koornwinder and
Jasper Stokman) which is part of the Askey-Bateman project. This version:
essential correction of equation (1.2.14) and some further minor correction
Quadratic transformations of Macdonald and Koornwinder polynomials
When one expands a Schur function in terms of the irreducible characters of
the symplectic (or orthogonal) group, the coefficient of the trivial character
is 0 unless the indexing partition has an appropriate form. A number of
q-analogues of this fact were conjectured in math.QA/0112035; the present paper
proves most of those conjectures, as well as some new identities suggested by
the proof technique. The proof involves showing that a nonsymmetric version of
the relevant integral is annihilated by a suitable ideal of the affine Hecke
algebra, and that any such annihilated functional satisfies the desired
vanishing property. This does not, however, give rise to vanishing identities
for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss
the required modification to these polynomials to support such results.Comment: 32 pages LaTeX, 10 xfig figure